Empty Set is Open in Metric Space

Theorem

Let $M = \struct {A, d}$ be a metric space.

Then the empty set $\O$ is an open set of $M$.

Proof

By definition, an open set $S \subseteq A$ is one where every point inside it is an element of an open ball contained entirely within that set.

That is, there are no points in $S$ which have an open ball some of whose elements are not in $S$.

As there are no elements in $\O$, the result follows vacuously.

$\blacksquare$